Removing the top case will allow you access to the GameCube's internal parts.
Turn over the Gamecube so that the bottom side is facing up.
Locate the four screws (circled in red) on each corner of the device. Then, use a 4.5 mm Gamebit screwdriver to remove all four screws.
With the bottom side of the GameCube facing upward and the screws removed, carefully pull the outer shell of the unit away from the top half. Then place the inside of the unit facing up.